In the case of the non-abelian \(SU(N)\)  Chern-Simons theory, the structure of the gauge orbits, which are associated with the \(SU(N)\)  connections, does not admit [1]} a simple description based on the homology group \(H_1(M)\) . Yet, in the characterization of the local degrees of freedom which are not related with the manifold topology, the non-abelian curvature \(F_A = 2 dA + i [A, A]\)  appears to play a fundamental role. In fact, the value \(F_A(x)\)  of the curvature in the point \(x\)  is specified [2]}, [3]} by the value of the gauge holonomy associated an infinitesimal loop centered in \(x\) , and each infinitesimal loop does not depend on the topology of \(M\) . Let us present a rough sketch of a possible argument that can be used to make this statement more precise.
